On mappings defined by G-methods and G-topological groups

Abstract

In this note,  the concepts of  $(G_{1},G_{2})$-open, $(G_{1},G_{2})$-closed, $(G_{1},G_{2})$-quotient and $(G_{1},G_{2})$--prefect mappings on arbitrary sets  are introduced and some theorems on them are established firstly. In particular, some results  improve the corresponding results in \cite{most14}. Secondly, we give a partial answer to the question posed by Li Liu \cite{liu19}. Finally, some properties of $G$-topological groups, $G$-connectedness and totally $G$-disconnectedness in $G$-topological groups are discussed.